A ratio is a way to compare two or more quantities by showing how many times one value is contained within another. It tells us the relative size of two or more values. Ratios are often used in problems involving mixtures, rates, proportions, and real-world comparisons.

For example, if you have 2 apples and 3 oranges, the ratio of apples to oranges is 2 : 3. It tells us that for every 2 apples, there are 3 oranges.

Basic Concepts of Ratios

• Writing Ratios: Ratios can be written in three different ways:
  • As a fraction: 2/3
  • With a colon: 2 : 3
  • In words: "2 to 3"
• Equivalent Ratios: Ratios can be scaled up or down by multiplying or dividing both terms by the same number. For instance, 2 : 3 is equivalent to 4 : 6 or 6 : 9.
• Simplifying Ratios: Ratios should always be expressed in their simplest form. To simplify a ratio, divide both terms by their greatest common divisor (GCD).

Methods for Solving Ratio Problems

1. Direct Relationships

When given a ratio and a value for one of the terms, you can find the missing value by scaling the ratio.

Example: If the ratio of boys to girls in a class is 2 : 3 and there are 10 boys, how many girls are there?

2x = 10 → x = 5

Girls = 3x = 3 × 5 = 15

2. Using Ratios to Solve Equations

Ratios are often used to set up equations. The key is to express all quantities using the same common value.

Example: If the sum of two numbers is 90, and the ratio is 2 : 3, find the numbers.

Let the two numbers be 2x and 3x.

Sum = 2x + 3x = 90 → 5x = 90 → x = 18

The two numbers are 2x = 36 and 3x = 54.

3. Solving Ratios Involving Percentages

Ratios can be combined with percentages. You often need to translate the percentage into a decimal or fraction to solve the problem.

Example: The ratio of 50% of one number to 25% of another number is given. Use the given information to set up an equation and solve.

4. Ratios Involving Geometry

Ratios are often used in geometry problems to find unknown sides, angles, or other measures.

Example: In a triangle, if the ratio of angles is 2 : 3 : 5, use the property that the sum of angles in a triangle is 180° to find each angle.

Let the angles be 2x, 3x, and 5x.

2x + 3x + 5x = 180 → 10x = 180 → x = 18

The angles are 36°, 54°, 90°.

Steps to Solve Ratio Problems

1. Identify the given quantities and relationships.

   Write down what each term represents and use a variable if necessary.

2. Set up the ratio.

   Use the ratio provided to represent the quantities in terms of a common variable (e.g., 2x, 3x).

3. Translate additional information into an equation.

   If the problem involves a sum, difference, or specific relationship, use that to create an equation.

4. Solve the equation for the variable.

   Solve for x and substitute back to find each term.

5. Check your answer.

   Make sure that the values satisfy the original ratio and any additional conditions given in the problem.

Example Problem and Solution

Problem: The ratio of two numbers is 3 : 5, and their sum is 64. What are the two numbers?

Solution:

1. Let the numbers be 3x and 5x.
2. According to the problem:

   3x + 5x = 64

3. Simplify the equation:

   8x = 64 → x = 8

4. Find the numbers:

   First number = 3x = 3 × 8 = 24

   Second number = 5x = 5 × 8 = 40

5. Answer: The two numbers are 24 and 40.